109 research outputs found
Low-Complexity LP Decoding of Nonbinary Linear Codes
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has
attracted much attention in the research community in the past few years. LP
decoding has been derived for binary and nonbinary linear codes. However, the
most important problem with LP decoding for both binary and nonbinary linear
codes is that the complexity of standard LP solvers such as the simplex
algorithm remains prohibitively large for codes of moderate to large block
length. To address this problem, two low-complexity LP (LCLP) decoding
algorithms for binary linear codes have been proposed by Vontobel and Koetter,
henceforth called the basic LCLP decoding algorithm and the subgradient LCLP
decoding algorithm.
In this paper, we generalize these LCLP decoding algorithms to nonbinary
linear codes. The computational complexity per iteration of the proposed
nonbinary LCLP decoding algorithms scales linearly with the block length of the
code. A modified BCJR algorithm for efficient check-node calculations in the
nonbinary basic LCLP decoding algorithm is also proposed, which has complexity
linear in the check node degree.
Several simulation results are presented for nonbinary LDPC codes defined
over Z_4, GF(4), and GF(8) using quaternary phase-shift keying and
8-phase-shift keying, respectively, over the AWGN channel. It is shown that for
some group-structured LDPC codes, the error-correcting performance of the
nonbinary LCLP decoding algorithms is similar to or better than that of the
min-sum decoding algorithm.Comment: To appear in IEEE Transactions on Communications, 201
Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance
Parameters of LDPC codes, such as minimum distance, stopping distance,
stopping redundancy, girth of the Tanner graph, and their influence on the
frame error rate performance of the BP, ML and near-ML decoding over a BEC and
an AWGN channel are studied. Both random and structured LDPC codes are
considered. In particular, the BP decoding is applied to the code parity-check
matrices with an increasing number of redundant rows, and the convergence of
the performance to that of the ML decoding is analyzed. A comparison of the
simulated BP, ML, and near-ML performance with the improved theoretical bounds
on the error probability based on the exact weight spectrum coefficients and
the exact stopping size spectrum coefficients is presented. It is observed that
decoding performance very close to the ML decoding performance can be achieved
with a relatively small number of redundant rows for some codes, for both the
BEC and the AWGN channels
Improved linear programming decoding of LDPC codes and bounds on the minimum and fractional distance
We examine LDPC codes decoded using linear programming (LP). Four
contributions to the LP framework are presented. First, a new method of
tightening the LP relaxation, and thus improving the LP decoder, is proposed.
Second, we present an algorithm which calculates a lower bound on the minimum
distance of a specific code. This algorithm exhibits complexity which scales
quadratically with the block length. Third, we propose a method to obtain a
tight lower bound on the fractional distance, also with quadratic complexity,
and thus less than previously-existing methods. Finally, we show how the
fundamental LP polytope for generalized LDPC codes and nonbinary LDPC codes can
be obtained.Comment: 17 pages, 8 figures, Submitted to IEEE Transactions on Information
Theor
A Unified Framework for Linear-Programming Based Communication Receivers
It is shown that a large class of communication systems which admit a
sum-product algorithm (SPA) based receiver also admit a corresponding
linear-programming (LP) based receiver. The two receivers have a relationship
defined by the local structure of the underlying graphical model, and are
inhibited by the same phenomenon, which we call 'pseudoconfigurations'. This
concept is a generalization of the concept of 'pseudocodewords' for linear
codes. It is proved that the LP receiver has the 'maximum likelihood
certificate' property, and that the receiver output is the lowest cost
pseudoconfiguration. Equivalence of graph-cover pseudoconfigurations and
linear-programming pseudoconfigurations is also proved. A concept of 'system
pseudodistance' is defined which generalizes the existing concept of
pseudodistance for binary and nonbinary linear codes. It is demonstrated how
the LP design technique may be applied to the problem of joint equalization and
decoding of coded transmissions over a frequency selective channel, and a
simulation-based analysis of the error events of the resulting LP receiver is
also provided. For this particular application, the proposed LP receiver is
shown to be competitive with other receivers, and to be capable of
outperforming turbo equalization in bit and frame error rate performance.Comment: 13 pages, 6 figures. To appear in the IEEE Transactions on
Communication
An Iterative Joint Linear-Programming Decoding of LDPC Codes and Finite-State Channels
In this paper, we introduce an efficient iterative solver for the joint
linear-programming (LP) decoding of low-density parity-check (LDPC) codes and
finite-state channels (FSCs). In particular, we extend the approach of
iterative approximate LP decoding, proposed by Vontobel and Koetter and
explored by Burshtein, to this problem. By taking advantage of the dual-domain
structure of the joint decoding LP, we obtain a convergent iterative algorithm
for joint LP decoding whose structure is similar to BCJR-based turbo
equalization (TE). The result is a joint iterative decoder whose complexity is
similar to TE but whose performance is similar to joint LP decoding. The main
advantage of this decoder is that it appears to provide the predictability of
joint LP decoding and superior performance with the computational complexity of
TE.Comment: To appear in Proc. IEEE ICC 2011, Kyoto, Japan, June 5-9, 201
Correcting a Fraction of Errors in Nonbinary Expander Codes with Linear Programming
A linear-programming decoder for \emph{nonbinary} expander codes is
presented. It is shown that the proposed decoder has the maximum-likelihood
certificate properties. It is also shown that this decoder corrects any pattern
of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where
\delta_A and \delta_B are the relative minimum distances of the constituent
codes).Comment: Part of this work was presented at the IEEE International Symposium
on Information Theory 2009, Seoul, Kore
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
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